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| + | To help answer this question, I consulted Signals and Systems, Oppenheim Willsky and Nawab. | ||
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Suppose we are given the following information about a signal x(t): | Suppose we are given the following information about a signal x(t): | ||
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2. x(t) is periodic with period T = 4 and Fourier coefficients <math> \ a_k </math>. | 2. x(t) is periodic with period T = 4 and Fourier coefficients <math> \ a_k </math>. | ||
| − | 3. | + | 3. <math> \ a_k = 0 </math> for <math> \left \vert k \right \vert > 1 </math>. |
| − | <math> \ a_k = 0 </math> for <math> \left \vert k \right \vert > 1 </math>. | + | |
| + | 4. <math> \frac{1}{2}\int_{0}^{2} \left \vert x(t) \right \vert ^2 \, dt = 1 </math>. | ||
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| + | Specify two different signals that satisfy these conditions. | ||
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| + | Two signals that would satisfy these coniditions is the input signal | ||
| + | |||
| + | <math> | ||
| + | \ x_1(t) = \sqrt{2} cos(2\pi t) | ||
| + | </math> | ||
| + | |||
| + | and | ||
| + | <math> | ||
| + | \ x_2(t) = - \sqrt{2} cos(2\pi t) | ||
| + | </math> | ||
| − | + | as cos(t) is an even function with a Fourier Series representation that has coefficients of 0 for absolute values of k greater than one. By multiplying the cosine function by the amplitude of the square root of 2 the signal provides the signal power of 1 unit when input into the power equation of specification (4). | |
Latest revision as of 18:33, 26 September 2008
To help answer this question, I consulted Signals and Systems, Oppenheim Willsky and Nawab.
Suppose we are given the following information about a signal x(t):
1. x(t) is real and even.
2. x(t) is periodic with period T = 4 and Fourier coefficients $ \ a_k $.
3. $ \ a_k = 0 $ for $ \left \vert k \right \vert > 1 $.
4. $ \frac{1}{2}\int_{0}^{2} \left \vert x(t) \right \vert ^2 \, dt = 1 $.
Specify two different signals that satisfy these conditions.
Two signals that would satisfy these coniditions is the input signal
$ \ x_1(t) = \sqrt{2} cos(2\pi t) $
and $ \ x_2(t) = - \sqrt{2} cos(2\pi t) $
as cos(t) is an even function with a Fourier Series representation that has coefficients of 0 for absolute values of k greater than one. By multiplying the cosine function by the amplitude of the square root of 2 the signal provides the signal power of 1 unit when input into the power equation of specification (4).
